The graph of the equation $y = |x| - 3$ is translated two units to the left and three units down. What are the coordinates of the minimum point of the new graph?
Since $|x|$ is nonnegative, it is minimized when it equals 0, which occurs when $x=0$.  So, the minimum point of the graph of $y=|x| - 3$ is $(0,-3)$.  When we translate this to the left two units and three units down, we get the point $\boxed{(-2,-6)}$.